The broad, long-term objective of this research project is to develop models and methodology for the analysis of repeated categorical responses. The repeated responses may be from a single outcome measured repeatedly through time, or they may be from several response variables measured one or more times each. It is often the case with repeated categorical responses that marginal distributions are of more interest than the complete joint distribution of the responses. The general theme of this proposal is likelihood based marginal modeling and its applications. A marginal model is specified in terms of a set of models for marginal distribution, and that set of models forms an implicit model for the joint distribution of responses. The approach to inference is a full likelihood approach, in the sense that the likelihood based on the joint distribution of the responses is what is maximized. The specific aims are to: (1) develop a stable algorithm for fitting marginal models by the method of maximum likelihood to large, sparse contingency tables; (2) development statistical methodology for marginal model based analyses in the presence of missing data; (3) explore marginal models within the context of exchangeable random variables; (4) develop parsimonious mixtures of marginal models that facilitate the estimation of scientifically interpretable parameters in the context of evaluating diagnostic tests; and (5) develop and evaluate profile likelihood methodology for marginal models. Multivariate categorical response data arise in many of the study designs used in biomedical research. In longitudinal studies a group of subjects is followed over time and data are typically collected at pre-specified points during the course of the study. In cross-over experiments subjects are randomized to one of several treatment sequence groups, wherein they receive a prescribed set of treatments in sequence. The successive treatment periods are usually separated by a suitably chosen period of time to allow the effects of the preceding treatments to washout of the subject's systems. In clinical trials there are frequently multiple endpoints of interest, such as response to therapy and side-effects of the therapy. The same is true for toxicological studies where there may be interest in, say, both birth status (e.g., normal, malformed, or dead) and birth weight (e.g., low, normal, high). In studies of diagnostic tests the observational units (e.g., a tissue sample) are routinely evaluated using a variety of tests and/or by a variety of evaluators (e.g., different laboratories, or pathologists). It is thus clear that statistical models and methodology for the analysis of repeated categorical responses are broadly applicable to a wide range of study designs frequently employed in health sciences research.